CIRCLES
CONVERTING FROM STANDARD FORM TO GENERAL FORM
OBJECTIVE
General Form of a Circle
So, we’ve already touched on the general form of a circle, but to recap, the general form of a circle looks like:
Ax 2 + By 2 + Cx + Dy + E = 0
So when do we use general form?
Well, honestly, most of the time we don’t, we actually try to convert it to standard form.
However there are many times when you’re working on a problem, and a general form of a circle may pop out.
In which case, you’re gonna wanna know that what you are working with is a circle.
Trust me, it gets annoying if you don’t see it.
So, let’s get started on converting!
Converting from General to Standard
So, let’s say you are given an equation that looks like:
x 2 + y 2 + 6x -10y + 29 = 0
The first step to solving is the same way we complete the square.
We want to subtract 29 from both sides
-29 -29
And now we are left with:
x 2 + y 2 + 6x -10y = -29
Now, let’s rearrange the equation so all of our x’s and y’s are together
x 2 + 6x + y 2 -10y = -29
Next, we complete the square for the x side:
x 2 + 6x
Remember, we take our b, divide it by 2a, and square it, then add it.
x 2 + 6x + 9 + y 2 -10y = -29 + 9
Now let’s complete the square for the y side:
y 2 -10y
Remember again, we take our b, divide it by 2a, and square it, then add it.
(x 2 + 6x + 9) + (y 2 -10y + 25) = -29 + 9 + 25
Now let’s simplify
And finally we are left with:
Ugh
Yeah I know, it’s a beast, but it is how we convert from general to standard.
So to do so, basically you complete the square for both the x and y variables, and simplify that way.
So, let’s do a few more just to make sure you understand how to do it.
(And yes, you will have homework/be quizzed/tested on this).
Example 1
So, let’s say you are given an equation that looks like:
x 2 + y 2 + 12x -40y + 100 = 0
The first step to solving is the same way we complete the square.
We want to subtract 100 from both sides
-100 -100
And now we are left with:
x 2 + y 2 + 12x -40y = -100
Now, let’s rearrange the equation so all of our x’s and y’s are together
x 2 + 12x + y 2 -40y = -100
Next, we complete the square for the x side:
x 2 + 12x
Remember, we take our b, divide it by 2a, and square it, then add it.
x 2 + 12x + 36 + y 2 -40y = -100 + 36
Now let’s complete the square for the y side:
y 2 -40y
Remember again, we take our b, divide it by 2a, and square it, then add it.
(x 2 + 12x + 36) + (y 2 -40y + 400) = -100 + 36 + 400
Now let’s simplify
And finally we are left with:
Example 2
So, let’s say you are given an equation that looks like:
x 2 + y 2 - 24x +20y - 75 = 0
The first step to solving is the same way we complete the square.
We want to add 75 to both sides
+75 +75
And now we are left with:
x 2 + y 2 - 24x +20y = 75
Now, let’s rearrange the equation so all of our x’s and y’s are together
x 2 - 24x + y 2 +20y = 75
Next, we complete the square for the x side:
x 2 - 24x
Remember, we take our b, divide it by 2a, and square it, then add it.
x 2 - 24x + 144 + y 2 +20y = 75 + 144
Now let’s complete the square for the y side:
y 2 +20y
Remember again, we take our b, divide it by 2a, and square it, then add it.
(x 2 - 24x + 144) + (y 2 +20y + 100) = 75 + 144 + 100
Now let’s simplify
And finally we are left with:
Example 3
So, let’s say you are given an equation that looks like:
x 2 + y 2 + 14x -18y - 10 = 0
The first step to solving is the same way we complete the square.
We want to subtract 100 from both sides
+10 +10
And now we are left with:
x 2 + y 2 + 14x -18y = 10
Now, let’s rearrange the equation so all of our x’s and y’s are together
x 2 + 14x + y 2 -18y = 10
Next, we complete the square for the x side:
x 2 + 14x
Remember, we take our b, divide it by 2a, and square it, then add it.
x 2 + 14x + 49 + y 2 -18y = 10 + 49
Now let’s complete the square for the y side:
y 2 -18y
Remember again, we take our b, divide it by 2a, and square it, then add it.
(x 2 + 14x + 49) + (y 2 -18y + 81) = 10 + 49 + 81
Now let’s simplify
And finally we are left with:
So how do we convert from General to Standard?�
Honestly, it’s way easier than you think.
To convert from general to standard, you just need to actually multiply the squares out, and combine like terms.
We’ll do this by using FOIL, but if you would like to use the area method to do this, that’s totally fine.
Convert the following:
So let’s say we have something like:
The first thing we are going to want to do is expand the parenthesis.
Now we distribute:
Now we distribute the y side:
Now combine them
Now combine like terms
Let’s make sure the equation equals 0:
-16 -16
Lastly make sure to rearrange so it looks like:
Ax 2 + By 2 + Cx + Dy + E = 0
And there it is!
AND THAT’S IT!
So even though it seems like a lot, it’s basically:
So, let’s try some more:
Example 1
So let’s say we have something like:
The first thing we are going to want to do is expand the parenthesis.
Now we distribute:
Now we distribute the y side:
Now combine them
Now combine like terms
Let’s make sure the equation equals 0:
-200 -200
Lastly make sure to rearrange so it looks like:
Ax 2 + By 2 + Cx + Dy + E = 0
And there it is!
Example 2
So let’s say we have something like:
The first thing we are going to want to do is expand the parenthesis.
Now we distribute:
Now we distribute the y side:
Now combine them
Now combine like terms
Let’s make sure the equation equals 0:
-64 -64
Lastly make sure to rearrange so it looks like:
Ax 2 + By 2 + Cx + Dy + E = 0
And there it is!
Example 3
Last one:
The first thing we are going to want to do is expand the parenthesis.
Now we distribute:
Now we distribute the y side:
Now combine them
Now combine like terms
Let’s make sure the equation equals 0:
-100 -100
Lastly make sure to rearrange so it looks like:
Ax 2 + By 2 + Cx + Dy + E = 0
And there it is!